XIONG ZHEN

(Department of Mathematics and Computer,Yichun University,Yichun,Jiangxi,336000)

Communicated by Du Xian-kun

1 Introduction

Consider the following equation given by

whereLis ann×nsymmetric real tridiagonal matrix,andBis the skew symmetric matrix obtained fromLby

whereL＞0(＜0)denotes the strictly upper(lower)triangular part ofL.In order to study the Toda lattice of statistical mechanics,the equation(1.1)was introduced by Flaschka[1],and this further was studied by Kodamaet al.[2],[3].

The notion of Hom-Lie algebras was introduced by Hartwiget al.[4]as part of a study of deformations of the Witt and the Virasoro algebras.In a Hom-Lie algebra,the Jacobi identity is twisted by a linear map,called the Hom-Jacobi identity.Someq-deformations of the Witt and the Virasoro algebras have the structure of a Hom-Lie algebra(see[4]and[5]).Because of close relation to discrete and deformed vector fields and differential calculus(see[4],[6]and[7]),more people pay special attention to this algebraic structure and their representations(see[8]and[9]).

We give an application of Hom-Lie algebras.Define a smooth map

whereR1is a subset ofR,s∈R1.In this paper,R1=R{0},orR1=R.

We mainly consider the following system:

wheres∈R1,andsis not dependent on variablet.[·,·]β(s)is just a Hom-Lie bracket,and(gl(V),[·,·]β(s),Adβ(s))is a Hom-Lie algebra(see[9]),where Adβ(s)(L)=β(s)Lβ(s)?1.

We study system(1.2)which is based on the following points:

(1)(1.2)is one parameter deformation of(1.1).Deformation theory is a very important field in singularity theory and bifurcation theory,and has many applications in science and engineering(see[10]and[11]).

(2)(1.2)is equivariant under the action of Lie group

This kind of differential equations is very important in equation theory and bifurcation theory(see[10]and[11]).

(3)For a Hom-Lie algebra(gl(V),[·,·]β(s),Adβ(s)),whenβ(s)=In,it is just a Lie algebra(gl(V),[·,·]),whereInis then×nidentity matrix.

The general framework is organized as follows:we first introduce the relevant definitions:one parameter deformation,Γ-equivariant and so on;then,we give definitions ofβ(s)and prove that{Adβ(s)|s∈R1}is a Lie group.Second,we give three kinds of one parameter deformation of(1.1).Then,we study these deformations respectively and give solutions.At last,some problems are given.

2 Preliminaries

We first give some definitions,one can find these definitions easily in[10]and[11].

Definition 2.1An equation

where x is an unknown variable,and the equation depends on an parameter s(∈R).For afixed s0,let g1(x)=g(x,s0).Then we call g(x,s)is one parameter deformation of g1(x).

Definition 2.2A smooth map g:Rn×R?→Rnis Γ-equivariant,if for any γ∈Γ,Γ is a Lie group,we have

where γx is the Lie group Γ action onRn.

Definition 2.3Let f,g:X?→Y be continuous maps.We say that f is homotopic to g if there exists a homotopy of f to g,that is,a map H:X×[0,1]?→Y such that H(x,0)=f(x)and H(x,1)=g(x).

We know that

Then we have

So,Gis a Lie group.

Proposition 2.1A map F1:R{0}?→G is given byF1(r)=Adβ(r),whereβ(r)=rIn.ThenF1is a homomorphism from a Lie group(R{0},×)to a Lie group G.

Proof.It is obvious that(R1,×)is a Lie group.We have

In particular,we have

Proposition 2.2A map F2:R?→G is given byF2(θ)=Adβ(θ),where

ThenF2is a homomorphism from a Lie group(R,+)to a Lie group G.

Proof.It is obvious that(R,+)is a Lie group.We have

The following is correct:

The proof is completed.

Remark 2.1DefineH:SO(n)×[0,1]?→SO(n)by

thenβ(θ)is homotopic toIn.Similarly,when

we haveβ1(θ)is homotopic toIn.Then

is homotopic to

So,we just study

Now,we let

We have

Obviously,by a direct calculation,we also have

Proposition 2.3With above notations,a map F3:R?→G is given byF3(λ)=Adβ(λ).

ThenF3is a homomorphism from a Lie group(R,+)to a Lie group G.

Proof.By a direct calculation,we have

At the same time,we have

Remark 2.2We can also defineH:SO(n)×[0,1]?→SO(n)by

Thenβ(λ)is homotopic toIn.

3 Main Results

3.1 Case of β(r)

In this case,(1.2)has the following form:

We have the following theorem.

So,in this case,deformation system(3.1)does not change properties of the solution of system(1.1).

3.2 Case of β(θ)

We just considern=2.Let

wherea,b,care functions which are independent of variablet.Then system(1.1)can be written as the following form:

System(1.2)has the following form:

By a direct calculation,we have the following results.

Theorem 3.2System(3.2)is integrable.then

is a solution of system(3.2).

3.3 Case of β(λ)

In this case,we just considern=2.By a direct calculation,the system(1.2)has the following form:

Then,we have the following facts.

Proposition 3.2

From the above discussion,we have the following theorem.

Theorem 3.3The system(3.3)is integrable if and only if λ=0.

So,the parameterλchange the integrability of system(1.1).

4 Problems

Whenn=3,we let

Therefore,the system(1.2)has the following form:

So,is this system(4.1)integrable?Whenn＞3,system(1.2)is integrable?

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